To generate a Pythagorean triple using the polynomial identity \( (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 \), we will use \( x = 7 \) and \( y = 3 \).
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First, calculate \( x^2 \) and \( y^2 \):
\[ x^2 = 7^2 = 49 \] \[ y^2 = 3^2 = 9 \]
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Next, compute \( x^2 + y^2 \) and \( x^2 - y^2 \):
\[ x^2 + y^2 = 49 + 9 = 58 \] \[ x^2 - y^2 = 49 - 9 = 40 \]
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Now compute \( 2xy \):
\[ 2xy = 2 \cdot 7 \cdot 3 = 42 \]
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According to the identity, we can now calculate the squares of these expressions:
\[ (x^2 + y^2)^2 = 58^2 = 3364 \] \[ (x^2 - y^2)^2 = 40^2 = 1600 \] \[ (2xy)^2 = 42^2 = 1764 \]
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By the polynomial identity, we have
\[ (x^2 + y^2)^2 = (x^2 - y^2)^2 + (2xy)^2 \]
\[ 3364 = 1600 + 1764 \]
This shows that the values \( (x^2 - y^2) \) and \( 2xy \) give us a Pythagorean triple:
- \( a = x^2 - y^2 = 40 \)
- \( b = 2xy = 42 \)
- \( c = x^2 + y^2 = 58 \)
Thus, the Pythagorean triple is \( (40, 42, 58) \).
Among the given options, 40 is one of the values of the Pythagorean triple.